Many biological and neural systems can be seen as networks of interacting periodic processes. Importantly, their functionality, i.e., whether these networks can perform their function or not, depends on the emerging collective dynamics of the network. Synchrony of oscillations is one of the most prominent examples of such collective behavior and has been associated both with function and dysfunction. Understanding how network structure and interactions, as well as the microscopic properties of individual units, shape the emerging collective dynamics is critical to find factors that lead to malfunction. However, many biological systems such as the brain consist of a large number of dynamical units. Hence, their analysis has either relied on simplified heuristic models on a coarse scale, or the analysis comes at a huge computational cost. Here we review recently introduced approaches, known as the Ott–Antonsen and Watanabe–Strogatz reductions, allowing one to simplify the analysis by bridging small and large scales. Thus, reduced model equations are obtained that exactly describe the collective dynamics for each subpopulation in the oscillator network via few collective variables only. The resulting equations are next-generation models: Rather than being heuristic, they exactly link microscopic and macroscopic descriptions and therefore accurately capture microscopic properties of the underlying system. At the same time, they are sufficiently simple to analyze without great computational effort. In the last decade, these reduction methods have become instrumental in understanding how network structure and interactions shape the collective dynamics and the emergence of synchrony. We review this progress based on concrete examples and outline possible limitations. Finally, we discuss how linking the reduced models with experimental data can guide the way towards the development of new treatment approaches, for example, for neurological disease.

中文翻译：

---

通过精确的平均场缩减了解生物和神经振荡器网络的动力学：综述。

许多生物和神经系统可以看作是相互作用的周期性过程的网络。重要的是，它们的功能，即这些网络是否可以执行其功能，取决于网络的新兴集体动力。振荡的同步是这种集体行为的最突出的例子之一，并且已经与功能和功能障碍相关联。了解网络结构和相互作用以及各个单元的微观特性如何塑造新兴的集体动力，对于发现导致故障的因素至关重要。但是，许多生物系统（例如大脑）由大量动力学单元组成。因此，他们的分析要么依赖粗略的简化启发式模型，要么分析将花费巨大的计算成本。在这里，我们回顾了最近引入的方法，称为Ott-Antonsen和Watanabe-Strogatz归约法，使一种方法可以通过将小比例尺和大比例尺桥接来简化分析。因此，获得了简化的模型方程，该方程仅通过少量的集体变量精确描述了振荡器网络中每个子种群的集体动力学。生成的方程式是下一代模型：它们不是启发式的，而是精确地链接了微观和宏观描述，因此可以准确地捕获基础系统的微观特性。同时，它们非常简单，无需大量计算即可进行分析。在过去的十年中，这些减少方法已成为了解网络结构和相互作用如何影响集体动力和同步性发展的工具。我们将基于具体示例回顾这一进展，并概述可能的局限性。最后，我们讨论将简化模型与实验数据联系起来如何指导开发新的治疗方法的方法，例如神经疾病。